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Extraction of nonlinear dynamics from short and noisy time series. (English) Zbl 0980.37034
Summary: We define efficient strategies for the modeling and predicting of short and noisy time series with neural networks. Several complementary methods are tested on short series constructed from the Lorenz system which has been spoiled with various levels of measurement or dynamical noise. The best strategies are selected from the simulation results according to the level and the noise characteristics. In the presence of measurement noise we show that over-sizing of the embedding dimension of the learning set increases the powerness of neural network fits. In the case of dynamical noise spoiling, we found that generation of a new trajectory predicted with local operators, amplifying information of the original series, allows the usage of neural networks as in the case of measurement noise, and so, avoids over-fitting problems possible with very short series. The strategies applied to the real biological and astronomical data (whooping cough in Great Britain and Wolf sunspot numbers) revealed their deterministic skeletons showing chaotic attractors.

MSC:
37M10 Time series analysis of dynamical systems
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[1] Casdagli, M; Eubank, S, Nonlinear modeling and forecasting, (1992), Addison-Wesley Reading
[2] Weigend, A.S; Gerhenfeld, N.A, Serie prediction: forecasting the future and understanding the past, (1993)), Addison-Wesley Reading
[3] Abarbanel, H.D.I, Analysis of observed chaotic data, (1996), Springer New York · Zbl 0875.70114
[4] Kantz, H; Schreiber, T, Nonlinear time series analysis, (1997), Cambridge University Press Cambridge · Zbl 0873.62085
[5] Cutler, C.D; Kaplan, D.T, Nonlinear dynamics and time series: building a bridge between the natural and statistical sciences, (1997), American Mathematical Society, Providence RI · Zbl 0855.00024
[6] Takens F. Detecting strange attractors in turbulence. In: Rand DA, Young L-S, editors. Dynamical systems and turbulence, vol. 898 (Warwick 1980) (Lecture Notes in Mathematics). Berlin: Springer; 1981. p. 366-81
[7] Abarbanel, H.D.I; Brown, R; Sidorowich, J.L; Tsimring, L.Sh, The analysis of observed chaotic data in physical systems, Rev mod phys, 65, 1331-1392, (1993)
[8] Sauer, T; Yorke, J.A; Casdagli, M, Embedology, J stat phys, 65, 579-616, (1991) · Zbl 0943.37506
[9] Kennel, M.B; Brown, R; Abarbanel, H.D.I, Determining embedding dimension for phase-space reconstruction using a geometrical construction, Phys rev A, 45, 3403-3411, (1992)
[10] Casdagli, M; Eubank, S; Farmer, J.D; Gibson, J, State space reconstruction in the presence of noise, Physica D, 51, 52-98, (1991) · Zbl 0736.62075
[11] Rosenstein, M.T; Collins, J.J; De Luca, C.J, Reconstruction expansion as a geometry-based framework for choosing proper delay times, Physica D, 73, 82-98, (1994)
[12] Fraser, A.M; Swinney, H.L, Independent coordinates for strange attractors from mutual information, Phys rev A, 33, 2, 1134-1140, (1986) · Zbl 1184.37027
[13] Theiler, J; Eubank, S; Longtin, A; Galdrikian, B; Farmer, J.D, Testing for nonlinearity in time series: the method of surrogate data, Physica D, 58, 77-94, (1992) · Zbl 1194.37144
[14] Chang, T; Sauer, T; Schiff, S.J, Test for nonlinearity in short stationary time series, Chaos, solitons & fractals, 5, 1, 118-126, (1995)
[15] Theiler, J; Eubank, S, Don’t bleach chaotic data, Chaos, solitons & fractals, 3, 4, 771-782, (1993)
[16] Schreiber, T; Grassberger, P, A simple noise-reduction method for real data, Phys lett A, 160, 411-418, (1991)
[17] Eckmann, J.-P; Ruelle, D, Ergodic theory of chaos and strange attractors, Rev mod phys, 57, 617-656, (1985) · Zbl 0989.37516
[18] Farmer, J.D; Sidorowich, J.J, Predicting chaotic time series, Phys rev lett, 59, 845-848, (1987)
[19] Casdagli, M, Nonlinear prediction of chaotic time series, Physica D, 35, 335-356, (1989) · Zbl 0671.62099
[20] Aguirre, L.A; Billings, S.A, Retrieving dynamical invariants from chaotic data using narmax models, Chaos, solitons & fractals, 5, 2, 449-474, (1995) · Zbl 0886.58100
[21] Boudjema, G; Mestivier, D; Cazelles, B; Chau, N.P, Using some recent techniques from chaos theory to analyse time-series in ecology, J biol syst, 3, 2, 291-302, (1995)
[22] Hornik, K; Stinchcombe, M; White, A, Multilayer feedforward networks are universal approximators, Neural networks, 2, 359-366, (1989) · Zbl 1383.92015
[23] Anderson, A.J; Rosenfeld, E, Neurocomputing: foundation of research, (1988), MIT press Cambridge
[24] Hertz, J; Krogh, A; Palmer, R.G, Introduction to the theory of neural computation, (1991), Addison-Wesley Reading
[25] Albano, A.M; Passamante, A; Hediger, T; Farrell, E, Using neural nets to look for chaos, Physica D, 58, 1-9, (1992) · Zbl 1194.68184
[26] Ghrist, R.W; Holmes, P.J, An ODE whose solutions contain all knots and links, Int J bifurcation chaos, 6, 5, 779-800, (1996) · Zbl 0878.34038
[27] Rulkov, N.F; Sushchik, M.M; Tsimring, L.S; Abarbanel, H.D.I, Generalized synchronization of chaos in directionally coupled chaotic systems, Phys rev E, 51, 2, 980-994, (1995)
[28] Flemming DM, Norbury CA, Crombie DL. Annual and seasonal variations in the incidence of common diseases. The Royal college of general practitioners; 1991
[29] Carbonell, M; Oliver, R; Ballester, J.L, A search for chaotic behaviour in solar activity, Astron astrophys, 290, 983-994, (1994)
[30] Jones, C.A; Weiss, N.O; Cattaneo, F, Nonlinear dynamos: a complex generalization of the Lorenz equations, Physica D, 14, 161-176, (1985) · Zbl 0579.76051
[31] Weiss, N.O, J stat phys, 39, 477-491, (1985)
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